Adyar – Adambakkam – Pallavaram – Pammal – Chromepet Now also at SELAIYUR Day - wise Portions Day 1 : Matrices and Determinants , Complex Numbers and Vector Algebra Day 2 : Analytical Geometry , Discrete Maths , Groups and Probability, Special 1 mark Test Day 3 : Differential Calculus, Applications of Integral Calculus, Differential Equations Centres at Pallavaram – Opp. St. therasa’s School Pammal – Krishna Nagar Adyar – Kasturiba Nagar Chrompet - Opp. MIT Selaiyur – Near Camp Road Junction Ph : 98404 48693 Prakash Institute Pallavaram - Opp. St. Therasa’s School Pammal – Krishna Nagar Chromepet – Opp. MIT college Selaiyur – Camp Road Adambakkam – DAV school Ph : 98404 48693 and 04465154818 Target Centum Practising Package +2 GENERAL MATHEMATICS All Practice papers is for 100 marks Duration to finish the paper is 1 Hr 30 Minutes Students should strictly complete the day wise syllabus Applications of Matrices & Determinants , Complex Numbers Section – A Choose the best option : 10 x 1 = 10 1. If A is a scalar matrix with scalar k 0, of order 3, then A – 1 is (a) ( 1 / k 2 ) I (b) ( 1 / k 3 ) I (c) ( 1 / k ) I (d) k I 2. If the matrix – 1 3 2 has an inverse then the value of k is 1 k – 3 1 4 5 (a) – 4 (b) 4 (c) 4 (d) – 4 3. If A is a 1x3 matrix then the rank of AAT is (a) 2 (b) 3 (c) 1 (d) 0 4. If A is a square matrix of order n then adj A is (a) A 2 (b) A n (c) A n – 1 (d) A 5. If A and B are any two matrices such that AB = O and A is non-singular then (a) B = O (b) B is singular (c) B is non – singular (d) B = A 6. The equations – 2x + y + z = l , x – 2y + z = m and x + y – 2z = n when l + m + n = 0 has (a) trivial solution (b) infinitely many solution (c) No solution (d) a non-zero unique solution 7. The inverse of the matrix 3 1 is 5 2 (a) 2 – 1 (b) – 2 5 (c) 3 – 1 (d) – 3 5 – 5 3 1 – 3 – 5 – 3 1 – 2 8. The rank of a diagonal matrix of order n is (a) 1 (b) 0 (c) n (d) n 2 Prakash Institute Pallavaram - Opp. St. Therasa’s School Pammal – Krishna Nagar Chromepet – Opp. MIT college Selaiyur – Camp Road Adambakkam – DAV school Ph : 98404 48693 and 04465154818 9. The system of equations ax + y + z = 0 , x + by + z = 0 and x + y + cz = 0 has a non – trivial solution then 1 / ( 1 – a ) + 1 / ( 1 – b ) + 1 / ( 1 – c ) = (a) 1 (b) 2 (c) – 1 (d) 0 10. In a system of 3 linear non – homogenous equations with three unknowns , if = 0 and x = 0 , y 0 and z = 0 then the system has (a) unique solutions (b) two solutions (c) no solution (d) infinitely many solutions 11. The value of [ ( – 1 + i 3 ) / 2 ] 100 + [ ( – 1 – i 3 ) / 2 ] 100 is (a) 2 (b) 0 (c) – 1 (d) 1 12. If x 2 + y 2 = 1 then the value of ( 1 + x + iy ) / ( 1 + x – iy ) is (a) x – iy (b) 2x (c) – 2iy (d) x + iy 13. If z represents a complex number then arg ( z ) + arg ( z ) is (a) /4 (b) / 2 (c) (d) 0 14. The value of i + i 22 + i 23 + i 24 + i 25 is (a) i (b) – i (c) 1 (d) – 1 15. The equation having 4 – 3i and 4 + 3i as roots is (a) x 2 + 8x + 25 = 0 (b) x 2 + 8x – 25 = 0 (c) x 2 – 8x + 25 = 0 (d) none 16. If is a cube root of unity then the value of ( 1 – + 2 ) 4 + ( 1 + – 2 ) 4 is (a) 0 (b) 32 (c) – 16 (d) – 32 17. If ( m – 5 ) + i ( n + 4 ) is the complex conjugate of (2m + 3 ) + i (3n – 2 ) then ( n , m ) is (a) ( – ½ , – 8 ) (b) ( – ½ , 8 ) (c) ( ½ , – 8 ) (d) ( ½ , 8 ) 18. If is the cube root of unity then the value of ( 1 – ) ( 1 – 2 ) ( 1 – 4 ) ( 1 – 8 ) is (a) 9 (b) – 9 (c) 16 (d) 32 19. If – z lies in the third quadrant then z lies in the (a) 1st quadrant (b) 2 nd quadrant (c) 3 rd quadrant (d) 4th quadrant 20. The least positive integer n such that [ ( 1 + i ) / ( 1 – i ) ] n = 1 is (a) 1 (b) 2 (c) 4 (d) 8 Section – B Answer any 5 questions from the following : 5 x 6 = 30 21. Find the inverse of the matrix 8 – 1 – 3 – 5 1 2 10 – 1 – 4 22. If A = 1 2 and B = 0 – 1 verify that ( AB ) – 1 = B – 1 A – 1 . 1 1 1 2 23. Find the rank of the matrix 4 2 1 3 6 3 4 7 2 1 0 1 24. Find the adjoint of the matrix A = 1 2 and verify A ( adj A ) = ( adj A ) A = AI. 3 – 5 25. Solve by matrix inversion method : 2x – y = 7 , 3x – 2y = 11. 26. P represents the variable complex number z, find the locus of P if arg [ ( z – 1 ) / ( z + 1 ) ] = / 3. Prakash Institute Pallavaram - Opp. St. Therasa’s School Pammal – Krishna Nagar Chromepet – Opp. MIT college Selaiyur – Camp Road Adambakkam – DAV school Ph : 98404 48693 and 04465154818 27. Prove that the points representing the complex numbers 2i , 1 + i , 4 + 4i and 3 + 5i on the Argand diagram are the vertices of a rectangle. 28. Find the square root of – 7 + 24i. 29. Solve the equation x 4 – 8x 3 + 24 x 2 – 32 x + 20 = 0 if 3 + i is a root. Section – C Answer any 5 questions from the following : 5 x 10 = 50 30. Solve using matrix inversion method : 2x – y + z = 7 , 3x + y – 5z = 13 , x + y + z = 5 31. A bag contains 3 types of coins namely Re.1, Rs.2 and Rs.5. There are 30 coins amounting to Rs.100 in total. Find the number of coins in each category. 32. Verify whether the given system of equations is consistent. If it is consistent , solve them. 2x + 5y + 7z = 52 , x + y + z = 9 , 2x + y – z = 0 33. Investigate for what values of , the simultaneous equations x + y + z = 6 , x + 2y + 3z = 10 , x + 2y + z = have (i) no solution (ii) unique solution (iii) an infinite number of solutions. 34. For what values of k , the system of equations kx + y + z = 1 , x + ky + z = 1 , x + y + kz = 1 have (i) unique solution (ii) more than one solution (iii) no solution 35. Solve x 9 + x 5 – x 4 – 1 = 0. 36. If n is a positive integer , prove that ( 3 + i ) n + ( 3 – i ) n = 2 n + 1 cos ( n / 6 ). 37. If and are the roots of x 2 – 2x + 2 = 0 and cot = y + 1 show that [ ( y + ) n – ( y + ) n ] / ( – ) = sin n / sin n . 38. If x = cos + i sin and y = cos + i sin , prove that x m y n + 1/x m y n = 2 cos ( m + n ). VECTOR ALGEBRA Section – A Choose the best option : 20 x 1 = 20 1. If a + b + c = 0 , a = 3,b = 4, c = 5 then the angle between a and b is (a) / 6 (b) 2 / 3 (c) 5 / 3 (d) / 2 2. The vectors 2 i + 3 j + 4 k and a i + b j + c k are perpendicular when (a) a = 2 , b = 3 , c = – 4 (b) a = 4 , b = 4 , c = 5 (c) a = 4 , b = 4 , c = – 5 (d) a = – 2 , b = 3 , c = 4 3. If the projection of a on b and projection of b on a are equal then the angle between a + b and a – b is (a) / 2 (b) / 3 (c) / 4 (d) 2 / 3 4. a and b are two unit vectors, is the angle between them, then ( a + b ) is a unit vector if (a) = / 3 (b) = / 4 (c) = / 2 (d) = 2 / 3 5. If a is a non-zero vector and m is a non-zero scalar then m a is a unit vector if (a) m = 1 (b) m = a (c) a = 1 / m (d) a = 1 6. If a and b include an angle 120 and their magnitude are 2 and 3 then a . b is (a) 3 (b) – 3 (c) 2 (d) – 3 / 2 7. If a + b = a – b , then (a) a is parallel to b (b) a is r to b (c) a = b (d) none Prakash Institute Pallavaram - Opp. St. Therasa’s School Pammal – Krishna Nagar Chromepet – Opp. MIT college Selaiyur – Camp Road Adambakkam – DAV school Ph : 98404 48693 and 04465154818 8. The work done by the force of magnitude 5 and direction along 2 i + 3 j + 6 k acting on a particle which is displaced from ( 4 , 4 , 4 ) to ( 5 , 5 , 5 ) is (a) 11 (b) 55 (c) 55 / 7 (d) 11 / 7 9. If a b = 0 where a and b are non zero vectors then (a) = 90 (b) = 45 (c) = 60 (d) = 0 10. If a b = a . b then (a) = 90 (b) = 45 (c) = 60 (d) = 0 11. If a = 3 , b = 4 and a . b = 9 then the value of a b is (a) 3 / 4 (b) 14 (c) 63 (d) 43 12. If a = 2 , b = 7 and a b = 3 i – 2 j + 6 k then (a) = / 4 (b) = / 6 (c) = / 2 (d) = / 3 13. If [ a x b , b x c , c x a ] = 64 then [ a , b , c ] is (a) 32 (b) 8 (c) 128 (d) 0 14.